System and method to facilitate pattern recognition by deformable matching

ABSTRACT

A system and method to facilitate pattern recognition or matching between patterns are disclosed that is substantially invariant to small transformations. A substantially smooth deformation field is applied to a derivative of a first pattern and a resulting deformation component is added to the first pattern to derive a first deformed pattern. An indication of similarity between the first pattern and a second pattern may be determined by minimizing the distance between the first deformed pattern and the second pattern with respect to deformation coefficients associated with each deformed pattern. The foregoing minimization provides a system (e.g., linear) that may be solved with standard methods.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.10/867,255, entitled, “SYSTEM AND METHOD TO FACILITATE PATTERNRECOGNITION BY DEFORMABLE MATCHING”, filed on Jun. 14, 2004 now U.S.Pat. No. 6,993,189, which is a continuation of U.S. patent applicationSer. No. 09/748,564 entitled, “SYSTEM AND METHOD TO FACILITATE PATTERNRECOGNITION BY DEFORMABLE MATCHING”, filed on Dec. 22, 2000, which isnow U.S. Pat. No. 6,785,419. The entireties of the aforementionedapplications are incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to pattern recognition and, moreparticularly, to a system and method to facilitate pattern recognitionby deformable matching.

BACKGROUND OF THE INVENTION

Pattern recognition systems are employed in various areas of technologyto help classify and/or match a test pattern relative to one or moreknown prototype patterns. Examples of pattern recognition applicationsinclude image analysis and classification, handwriting recognition,speech recognition, man and machine diagnostics, industrial inspection,medical imaging, etc.

In a pattern recognition system, it is common to store large amounts ofdata indicative of prototype patterns and compare them to a givenexample or unknown input symbol for identification. Several commonalgorithms may be utilized, such as K-nearest Neighbor (KNN), Parzenwindows, and radial basis function (RBF), to compare or measuresimilarities between patterns. A level of similarity may be determinedby generating a distance measure. By way of example, a simple algorithmfor comparing two patterns f and g is to compute the Euclidean distancebetween them, such as may be expressed as:

${d_{E}( {f,g} )} = {\sqrt{\sum\limits_{x,y}\;( {{f( {x,y} )} - {g( {x,y} )}} )^{2}} = \sqrt{( {f - g} )^{2}}}$where d_(E) denotes the Euclidean distance between two patterns, and fand g are assumed to be two-dimensional patterns, indexed by x and y. Anextension of the Euclidean distance methodology to other dimensions isstraightforward.

The usefulness of the Euclidean distance algorithm is limited, however,because if f and g are not perfectly aligned, the Euclidean distance canyield arbitrarily large values. Consider, for instance, a case where gis a translated version of f, that is g(x,y)=f(x+1, y). In this case,the Euclidean distance could yield a very large value, even thought andg may be virtually identical except for a small amount of translation inthe x-direction.

One proposed comparison scheme to remedy the aforementioned shortcomingassociated with the traditional Euclidean distance approach is to employa tangent distance, such as is disclosed in U.S. Pat. No. 5,422,961.This comparison scheme is invariant with respect to a selected set ofsmall transformations of the prototypes. The small transformations ofinterest are expressed by calculating a derivative of the transformedimage with respect to the parameter that controls the transformation.The directional derivative is used to generate a computationalrepresentation of the transformation of interest. The transformation ofinterest, which corresponds to a desired invariance, can be efficientlyexpressed by using tangent vectors constructed relative to the surfaceof transformation. The tangent distance d_(T) may be expressed as:

${d_{T}( {f,g} )}^{2} = {\min\limits_{\alpha_{f}\alpha_{g}}( {f + {L_{f}a_{f}} - g - {L_{g}a_{g}}} )^{2}}$where L_(f) and L_(g) are matrices of tangent vectors for f and grespectively, and a_(f) and a_(g) are vectors representing the amount ofdeformation along the tangent plane. An advantage of tangent distancecompared to the traditional Euclidean distance approach is that thetangent distance is less affected by translation than the Euclideandistance because if L_(f) and L_(g) contain a linear approximation ofthe translation transformation, the tangent distance compares thetranslated version of f and g. The tangent distance concept is exploredin greater detail in a paper entitled, “Efficient Pattern RecognitionUsing a New Transformation Distance,” presented by Patrice Y. Simard,Yann LeCun and John Denker, Advances in Neural Information ProcessingSystems, Eds. Morgan Kaufmann, pp. 50-58, 1993.

A limitation of tangent distance approach, however, is that thetransformations to which it is invariant generally must be knowna-priori and precisely (e.g., translation, rotation, scaling, etc.).Moreover, tangent distance has no mechanism to specify loose constraintssuch as small elastic displacements. Such mechanism would be usefulbecause in many cases, such as with speech or image patterns, it is notknown which transformations should be used, but it is assumed that theproblem exhibit some invariance with respect to small elasticdisplacements.

A comparison scheme that is invariant to such transformations canoperate with greater economy and speed than comparison schemes thatrequire exhaustive sets of prototypes. By way of example,transformations of alphanumeric patterns that are of interest in thisregard may include translation, rotation, scaling, hyperbolicdeformations, line thickness changes, and gray-level changes. Anydesired number of possible invariances can be included in any particularrecognition process, provided that such invariances are known a priori,which is not always possible.

Many computer vision and image processing tasks benefit from invariancesto spatial deformations in the image. Examples include handwrittencharacter recognition, face recognition and motion estimation in videosequences. When the input images are subjected to possibly largetransformations from a known finite set of transformations (e.g.,translations in images), it is possible to model the transformationsusing a discrete latent variable and perform transformation-invariantclustering and dimensionality reduction using Expectation Maximizationas in “Topographic transformation as a discrete latent variable” byJojic and Frey presented at Neural Information Processing Systems (NIPS)1999. Although this method produces excellent results on practicalproblems, the amount of computation grows linearly with the total numberof possible transformations in the input.

A tangent-based construction of a deformation field may be used to modellarge deformations in an approximate manner. The tangent approximationcan also be included in generative models, such as including linearfactor analyzer models and nonlinear generative models. Another approachto modeling small deformations is to jointly cluster the data and learna locally linear deformation model for each cluster, e.g., using EM in afactor analyzer. With the factor analysis approach, however, a largeamount of data is needed to accurately model the deformations. Learningalso is susceptible to local optima that might confuse deformed datafrom one cluster with data from another cluster. That is, some factorstend to “erase” parts of the image and “draw” new parts, instead of justperturbing the image.

SUMMARY OF THE INVENTION

The present invention relates to a system and method to facilitatepattern recognition or matching between patterns. A deformation field ismultiplied with a spatial derivative of a first pattern and a resultingdeformation component is added to the first pattern to derive a firstdeformed pattern. The deformation field is generally non-uniform and maybe parameterized by a smooth functional basis having associateddeformation coefficients. By way of example, the functional basis maycorrespond to inverse wavelet transforms of varying frequency orresolution defined by the associated deformation coefficients. A minimumdistance between the first deformed pattern and a second pattern may bedetermined by minimizing the distance between the first deformed patternand the second pattern with respect to the deformation coefficients ofthe first deformed pattern. Advantageously, the foregoing minimizationprovides a system (e.g., linear) that may be solved with standardmethods, such as quadratic optimization.

In accordance with one aspect of the present invention, the secondpattern also may be deformed. A second deformed pattern is obtained bymultiplying a second deformation field with a spatial derivative of thesecond pattern and by adding the resulting deformation component to thesecond pattern. The second deformation field is generally non-uniformand may be parameterized by a second smooth functional basis havingassociated deformation coefficients. By way of example, the secondfunctional basis may correspond to inverse wavelet transforms of varyingfrequency or resolution defined by the associated deformationcoefficients. When the second pattern is deformed, the minimum distancebetween the first and second deformed patterns may be determined byminimizing the distance between the deformed patterns with respect totheir associated deformation coefficients. Advantageously, the foregoingis a standard quadratic optimization problem which reduces to solving alinear system of equations.

One aspect of the present invention provides a method to facilitatepattern recognition between a first pattern and a second pattern. Asubstantially smooth deformation field having deformation coefficientsis applied to the first pattern to deform the first pattern and form adeformation component of the first pattern. The first pattern and thedeformation component of the first pattern are aggregated to define afirst deformed pattern. A distance between the first deformed patternand the second pattern is determined as a function of the deformationcoefficients for the first pattern.

Another aspect of the present invention provides a method to facilitatepattern recognition between a first pattern and a second pattern. Asubstantially smooth deformation field having deformation coefficientsis applied to deform the first pattern and form a first deformationcomponent. The first deformation component is added to the first patternto define a first deformed pattern. The deformation coefficients for thefirst pattern are determined by minimizing a distance between the firstdeformed pattern and the second pattern. The determined deformationcoefficients may then be employed to evaluate a distance between thefirst deformed pattern and the second pattern.

In accordance with yet another aspect of the present invention, thesecond pattern also may be deformed by applying a second deformationfield to the second pattern to form a second deformation component. Thesecond deformation component is added to the second pattern to define asecond deformed pattern. The second deformation field is parameterizedby functional basis vectors and deformation coefficients associated withthe second pattern. A distance between the first deformed pattern andthe second deformed pattern may then be determined as a function of therespective deformation coefficients.

Still another aspect of the present invention provides a system tomeasure similarity between first and second patterns. The systemincludes a first deformation system for applying a substantially smoothdeformation field to deform the first pattern and provide a firstdeformation component. An aggregation system aggregates the firstdeformation component and the first pattern to derive a first deformedpattern. A minimization system is operable to minimize a distancebetween the first deformed pattern and the second pattern as a functionof the deformation field.

To the accomplishment of the foregoing and related ends, certainillustrative aspects of the invention are described herein in connectionwith the following description and the annexed drawings. These aspectsare indicative, however, of but a few of the various ways in which theprinciples of the invention may be employed and the present invention isintended to include all such aspects and their equivalents. Otheradvantages and novel features of the invention will become apparent fromthe following detailed description of the invention when considered inconjunction with the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram of a system that may be employed tomeasure similarity between patterns in accordance with the presentinvention;

FIG. 2 is a functional block diagram of another system that may beemployed to measure similarity between patterns in accordance with thepresent invention;

FIG. 3 is a functional block diagram of yet another system that may beemployed to measure similarity between patterns in accordance with thepresent invention;

FIG. 4 is a graphical depiction of a relationship of two patterns beingdeformably matched in accordance with the present invention;

FIG. 5 is a functional block diagram of an operating environment thatmay be employed to implement a process in accordance with the presentinvention;

FIG. 6A is a flow diagram illustrating a methodology for determiningdeformation coefficients in accordance with the present invention;

FIG. 6B is a flow diagram illustrating a methodology for deriving ameasure of similarity between patterns in accordance with the presentinvention;

FIG. 7 is an example of a pattern that has been deformed by multipledeformation field in accordance with the present invention; and

FIG. 8 is an example of two image patterns that have been deformablymatched in accordance with the present invention.

NOTATION

The present system can be used for patterns of arbitrary dimensionalitythat exhibit some type of spatial coherence during deformation. In otherwords, it is believed that the inherent dimensionality of thedeformation is substantially smaller than that of a pattern, due to thefact that the deformation cannot change considerably among theneighboring parts of the pattern. In case of images, for example, if thepart in a lower right corner is compressed, it is not likely that thepart right above it will be considerably expanded. The deformations inimages are usually smooth.

Having said that, it also should be noted that regardless of thedimensionality of the pattern, the notation is considerably simplifiedif the patterns are represented as vectors of values. In case of M×Nimages, for example, which are inherently two-dimensional, the imagescan be unwrapped into a long vector by stacking M-long image columns ontop of each other (in fact, this is how the matrices of anydimensionality are typically stored in memory). This will result in avector with M*N elements. We will denote the pattern vectors with boldface lower case letters, such as f and g. By way of illustration, incase of images, the number of elements in these vectors will be equal tothe total number of pixels in the image, and in case of speech/audiopatterns, the number of elements will be equal to the number of samplesor frames in the time window. This simplification in notation allowsdescribing operators acting on patterns of arbitrary dimensionality tobe characterized by two-dimensional matrices and the use of simplelinear algebra for derivations.

For example, in case of an image pattern f, an approximate derivative inY direction could be the matrix

$G_{y} = \begin{matrix}1 & 2 & {3\mspace{11mu}\ldots} & \; & \; & \; & M & {M + {1\mspace{11mu}\ldots}} & \; & \; & \; & {MN} & \; \\1 & {- 1} & 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\0 & 1 & {- 1} & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & \ldots & 0 & 2 \\\ldots & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \cdots \\0 & 0 & 0 & 0 & 0 & \ldots & 0 & 1 & {- 1} & 0 & \ldots & 0 & {M + 1} \\\ldots & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\0 & 0 & \ldots & \; & \; & \; & \; & \; & \; & \; & \; & 1 & {MN}\end{matrix}$The letters in the first row denote the column and the letters in thelast column denote the rows of the matrix. In this example, thederivative is approximated by a difference between the pixel and itsimmediate neighbor. This is accomplished by placing 1 and −1 in theappropriate places in each row. Since the image is represented as a longvector, care has to be taken at the M-th row of G since the appropriatepixel does not have the neighbor (edge of the image). Then a spatialderivative can be computed by multiplying the pattern f with the matrixG_(x).

When we take the derivative in the other direction, however, 1 and −1values have to be M entries apart since the neighbor in x direction isin the next column of an image, which is equivalent to being M entriesapart in the vector representation:

$G_{x} = \begin{matrix}1 & 2 & {3\mspace{11mu}\ldots} & \; & \; & \; & M & {M + {1\mspace{11mu}\ldots}} & \; & \; & \ldots & {MN} & \; & \; \\1 & 0 & 0 & 0 & 0 & \ldots & 0 & {- 1} & 0 & 0 & \ldots & 0 & 1 & \; \\0 & 1 & 0 & 0 & 0 & \ldots & 0 & 0 & {- 1} & 0 & \ldots & 0 & 2 & \; \\\ldots & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \cdots & \;_{\;} \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\0 & 0 & \ldots & \; & \; & \; & \; & \; & \; & \; & \; & 1 & {MN} & \;\end{matrix}$

Note, however, that other derivative approximations can be described insimilar way. It is to be understood and appreciated that the abovematrices have been provided for purposes of illustration only, and thescope of this application is not limited to these particularapproximations. Those skilled in the art will understand and appreciatethat other approximations could be utilized in accordance with an aspectof the present invention.

This illustrates that vector representation of the patterns allowslinear operations, even when they deal with different dimensions of thepattern, to be expressed by matrix multiplications and thus allvariables in our notation are at most two-dimensional.

To make it easier to track the dimensionality of different terms inequations presented in the following description, lower case lettersrepresent scalars (distance d, for example), bold-face lower-caseletters represent vectors (such as a pattern f, or a vector ofdeformation coefficients a) and bold-face capital letters representmatrices, such as the above difference operators G_(x) and G_(y).

DETAILED DESCRIPTION OF THE INVENTION

The present invention provides a new model that may be utilized tomeasure similarity between patterns to facilitate pattern recognition ormatching. The present invention employs a deformation field that is alinear combination of functional basis vectors one or more patterns. Thefunctional basis vectors may be represented as low frequency basisvectors, such as low-frequency wavelet basis vectors, which allowsignificantly fewer parameters to represent a wide range of realisticsmooth deformations. In this way, a pattern recognition system, inaccordance with the present invention, is less likely to use adeformation field to “erase” part of an image or “draw” a new part,because the field is usually not smooth.

In the following description, for purposes of explanation, certaindetails are set forth in order to provide a thorough understanding ofthe present invention. It will be evident to one skilled in the art,however, that the present invention may be practiced without thesespecific details. In other instances, well-known structures andfunctional aspects are shown and described in block diagram form inorder to facilitate description of the present invention. Those skilledin the art will understand and appreciate various ways to implement suchexamples, all of which are contemplated as being within the scope of thepresent invention.

Turning now to FIG. 1, a functional block diagram of a system 10 tofacilitate matching or recognizing patterns f and g, in accordance withan aspect of the present invention is, illustrated. The patternrecognition system 10 includes a deformation system 12 operable todeform the pattern f to derive a deformation component 14 thereof. Thedeformation component 14 is aggregated with the original pattern f toform a deformed version 16 of the pattern, indicated at f.

The deformation system 12, for example, applies a smooth, non-uniformdeformation field to a spatial derivative of the pattern f Thedeformation field may be parameterized by smooth, functional basisvectors and associated deformation coefficients. The extent of thedeformation coefficients determines the localization of the deformationbeing applied to the pattern f. That is, a larger number of coefficientsmay be utilized to implement more localized deformation of the pattern.

By way of example, the functional basis vectors may correspond toinverse wavelet transforms of varying frequency or resolution defined bythe associated deformation coefficients. Computing low frequency waveletcoefficients only, with all the higher frequency coefficients set tozero, may be utilized to mitigate computational processing associatedwith the recognition (or matching) process.

The system 10 also includes a distance determining system 18 forderiving an indication of the distance 20 between the deformed pattern16 and the other pattern g. By way of example, the pattern f may be atest pattern while the pattern g may be prototype pattern or vice versa.The distance between the deformed pattern {tilde over (f)} and thepattern g thus may be determined by subtracting the pattern g from thedeformed pattern {tilde over (f)} (e.g., d(f,g)=∥{tilde over (f)}−g ∥).

The system 10 also may include a cost function 22 that is applied to thedistance 20 to impose a cost (or penalty) and provide a weightedindication of the distance d′(f, g), indicated at 24. The cost function22, for example, may impose a cost that is functionally related to thedeformation coefficients (indicated by connection 26), such that, forexample, higher resolution deformations have a greater associated costthan low resolution deformations. Alternatively or additionally, a costmay be computed based on pattern g and/or the deformation component 14(indicated by connection 28) of the pattern f.

When the deformation coefficients are unknown, they may be computed as afunction of the distance d′({tilde over (f)},g) provided at 24 (oralternatively from the distance 20). That is, corresponding deformationcoefficients, for example, may be determined by minimizing ∥{tilde over(f)}−g∥ with respect to the deformation coefficients provided by thedeformation function 12. Another option is to compute the deformationcoefficients in an iterative manner (e.g., by gradient descent), such asbased on feedback 30 from the weighted distance 24 (or alternativelyfrom the distance 20) to the deformation function 12.

FIG. 2 illustrates another example of a system 50 that may be utilizedto measure the similarity between patterns f and g in accordance with anaspect of the present invention. In the system 50, both input patterns fand g are deformed. More particularly, deformation functions 52 and 54are employed to derive a deformation component for the respectivepatterns f and g, indicated respectively at 56 and 58. The deformationcomponents 56 and 58 are added to the respective, non-deformed patternsf and g to derive corresponding deformed patterns, indicatedrespectively at 60 and 62.

The respective deformation components 56 and 58 may be derived byapplying a deformation field having associated deformation coefficientsto a derivative of each respective pattern. For example, if the patternsare images in pixel space, and each of them can be deformed by adeformation field parameterized by X and Y coefficients, respectively,the two deformation components 56 and 58 can be obtained by multiplyingthose fields with the spatial derivatives of each of the patterns f andg. Those skilled in the art will understand and appreciate that thepatterns f and g may correspond to other types of patterns (e.g., speechpatterns, celestial patterns, stock market information, etc.) in acorresponding dimensional space.

Each deformation field may be parameterized by functional basis vectorsand associated deformation coefficients. The functional basis vectorsmay be derived from an inverse wavelet transform, a fast Fouriertransform, or other suitable transform (e.g., sin or cosine) operable toprovide a substantially smooth deformation field. An inverse wavelettransform is particularly desirable, as it may be utilized tocharacterize the deformation field with low frequency wavelets andassociated deformation coefficients. The deformation coefficients may beselected to capture global trends in the pattern(s) being recognized,such as using only the coefficient corresponding to very smoothdeformation. Alternatively, a greater number of coefficients may beutilized to capture more localized deformations.

By way of illustration, even though all deformation fields can berepresented with the full set of wavelet coefficients, a useful subsetof fields (e.g., the substantially smooth deformation fields) can berepresented with a small subset of the available coefficients. Adeformed image could be obtained by applying the deformation fielddirectly to the image, but this would be quite complicated because oneor more points likely would not land on an associated sampling grid, andbecause such operation is not linear.

In order to facilitate the computations associated with forming thedeformed image, a linear approximation of the deformation (representedby the deformation field) is made. The deformed image may then beobtained by multiplying the deformation field with a spatial derivativeof the image, and adding the result to the original image. An addedadvantage of the linear approximation is that the deformationcoefficients can be found by solving a linear system.

A distance determining system 64 is employed to derive a distance 66between the deformed patterns {tilde over (f)} and {tilde over (g)},such as d′(f, g)=∥{tilde over (f)}−g∥. The distance 66 provides aquantitative indication as to whether there is a match between thepatterns f and g.

The system 50 further may include a cost function 68 for imposing a costand/or weighting different aspects of the distance 66 or part of thesystem utilized to derive the distance 66. The cost function 68 providesa modified measure of the distance between the patterns, indicated at70, which reflects the cost imposed thereon. The cost function 68, forexample, may include a variant structure associated with the deformationcomponents 56 and 58 (indicated by connections 72 and 74). Additionallyor alternatively, the cost may include a covariant structurefunctionally related to the deformation coefficients of eachcorresponding deformation system 52 and 54 (indicated by connections 76and 78) and/or functionally related to the computed distance at 66.

In a situation where the deformation coefficients are unknown, they maybe computed by solving a linear system defined by the distance 66 or 70between the deformed images. By way of example, the minimization mayinclude determining the minimum of ∥{tilde over (f)}−{tilde over (g)}∥with respect to the deformation coefficients associated with thedeformed patterns. Advantageously, this may be solved by straightforwardmethods, such as quadratic optimization. Alternatively, feedback 80 maybe utilized as part of an iterative process (e.g., gradient descent) tocompute the respective deformation coefficients associated with eachdeformation function 52 and 54.

In accordance with an aspect of the present invention, the deformationcoefficients of the first pattern may be tied to the deformationcoefficient of the second pattern. For instance, assuming that bothpatterns f and g have identical displacement basis functions, thedeformation coefficients of the first pattern may be set to be theopposite of the deformation coefficient of the second pattern, thusreducing the size of the optimization problem and further constrainingthe allowable deformations.

FIG. 3 is a functional block diagram of a system 100 that may beemployed, in accordance with an aspect of the present invention, tomeasure the similarity between two patterns f and g. For purposes ofsimplicity of explanation, the patterns f and g will be described asbeing digital images, although it is to be understood and appreciatedthat the system 100 is equally applicable to other types of patterns.Those skilled in the art also will understand and appreciate that theprinciples described below may be extended to numerous applications,such as video coding, motion estimation, optical character recognitionand video mosaic, to name a few.

The image f is supplied to derivative functions 102 and 104, whichperform partial derivatives on the image f to provide respective spatialderivatives of the image with respect to x and y, indicated respectivelyat 106 and 108.

An x-component of a deformation field 110 is applied, such as by anelement-wise product function to the partial derivative 106 with respectto x to derive a corresponding deformation component 112 of the imagewith respect to x. Similarly, a y-component of a deformation field 114is applied to the partial derivative 108, with respect to y, via anelement-wise product function to derive a deformation component 116 ofthe image f The deformation field (δ_(x),δ_(y)), which is the aggregateof the field components 110 and 114, may be a non-uniform field that isparameterized by a smooth functional basis having associated deformationcoefficients. The functional basis may be in the form of vectors derivedfrom, for example, an inverse wavelet transform, a fast Fouriertransform, or other suitable transform (e.g., sin or cosine) operable toprovide a substantially smooth deformation field. The resultingdeformation components 112 and 116 of the image f are added together bya summing function 118 to form an aggregated deformation component 120for the image f. The resulting deformation component 120 is, in turn,added to the image f by a summing function 122 to derive a deformedimage {tilde over (f)}, indicated at 124. The deformed image {tilde over(f)} thus may be represented as

$\begin{matrix}{{\overset{\sim}{f} = {f + {\frac{\partial f}{\partial x} \circ \delta_{x}} + {\frac{\partial f}{\partial y} \circ \delta_{y}}}},} & ( {{Eq}.\mspace{14mu} 1} )\end{matrix}$

-   -   where “∘” denotes element-wise multiplication, sometimes in        mathematical software denoted by “.*”.

In accordance with an aspect of the present invention, the deformationfield (δ_(x), δ_(y)) is a substantially smooth deformation field isconstructed from low frequency wavelets basis vectors stored as columnsin matrix R. For example, an x-component R_(x) of a wavelet basis vectoris multiplied by associated deformation coefficients a_(x) to form thedeformation field component 110 and a y-component R_(y) of the waveletbasis vector is multiplied by deformation coefficients a_(y) to form thedeformation field component 114. As a result, the deformation field maybe expressed in its component parts asδ_(x)=R_(x)a_(x),  (Eq.2)δ_(y)=R_(y)a_(y),

-   -   where the columns of R contain low-frequency wavelet basis        vectors (R_(x), R_(y)), and

$a = \begin{bmatrix}a_{x} \\a_{y}\end{bmatrix}$are the deformation coefficients. It should be noted, that the number ofbasis functions is considerably smaller than the dimensionality of thevector describing the pattern (the total number of pixels in the image),and thus the deformation field, which consists of as many values as thepattern, may be described by a considerably smaller number ofdeformation coefficients The number of parameters in a_(x) and a_(y) canbe determined a priori, or by trial and error.

An advantage of wavelets is their space/frequency localization. Forexample, low-frequency deformation coefficients may be selected tocapture global trends in the pattern(s) being recognized, such as byutilizing a number of deformation coefficients that is a small fractionof the number of pixels in the image f. The deformations localized insmaller regions of the image can be expressed by more spatiallylocalized (e.g., higher frequency) wavelets. As a result, the deformedimage {tilde over (f)} may be represented as a linear approximation,which mitigates computation time, namely:{tilde over (f)}=f+(G _(x) f)∘(Ra _(x))+(G _(y) f)∘(Ra _(y)),  (Eq. 3)where the derivatives in Eq. 1 are approximated by sparse matrices G_(x)and G_(y) that operate on f to compute finite spatial differences in theimage pattern.

It is to be appreciated that the deformation field components 110, 114is decoupled from the image f. That is, the deformation component isdecomposed along its basis of displacement and, thus, the deformed imagemay be approximated by a linear transformation, as indicated in Eq. 3.As a result, a particular deformation field may be easily applied toother patterns to deform the pattern in a corresponding way.

It is also to be appreciated that Eq. 3 is bilinear in the deformationcoefficients (a_(x),a_(y)) and in the original image f. That is, Eq. 3is linear in f given a and it is linear in a given f. The element-wiseproduct thus may be rewritten as a matrix product, such as by convertingeither the vector Gf or the vector Ra to a diagonal matrix using a diag( ) function:{tilde over (f)}=f+D(f)a, where D(f)=[diag(G _(x) f)R _(x) diag(G _(x)f) R _(y)]  (Eq. 4){tilde over (f)}=T(a)f, where T(a)=[I+diag(R _(x) a _(x))G _(x)+diag (R_(y) a _(y))G _(y)].  (Eq. 5)

Eq. 4 seems like a typical linear model such as factor analysis orprincipal component analysis. It should be noted, however, that thecomponents in this case need not to be estimated, as they are based onthe deformation model D(f).

Eq. 4 shows by applying a simple pseudo inverse, the deformationcoefficients that transform f into {tilde over (f)} may be estimated:a=D(f)⁻¹({tilde over (f)}−f).  (Eq. 6)The low-dimensional vector of coefficients provided by Eq. 6 minimizesthe distance ∥f−{tilde over (f)}∥.

Given a test (or prototype) image g, the image f may be matched relativeto g by computing the deformation coefficients, a=D(f)⁻¹(g−f), thatminimize ∥f−g∥. In accordance with another aspect of the presentinvention the image g, which may be an N-dimensional vector of a digitalimage in the same space as the image f, also may be deformed. Thisenables more extreme deformations to be successfully matched inaccordance with the present invention.

Returning now to FIG. 3, the deformation of the image g may occur insubstantially the same manner as described above with respect to theimage f. Briefly stated, the image g is supplied to partial derivativefunctions 130 and 132, which perform partial derivatives on the image gto provide partial derivatives 134 and 136 of the image with respect tox and y, respectively. A deformation field (δ_(x),δ_(y)), isparameterized by a smooth functional basis having functional basisvector components R_(x) and R_(y) and associated deformationcoefficients b_(x) and b_(y).

By way of example, the deformation field is a substantially smoothdeformation field constructed from low frequency wavelets. Deformationcoefficients b_(x) and b_(y) are multiplied by wavelet basis vectorsR_(x) and R_(y) to form x and y-components 138 and 140 of thedeformation field, which may be expressed asδ_(x)=R_(x)b_(x),  (Eq. 7)δ_(y)=R_(y)b_(y),

-   -   where (R_(x), R_(y)) are the wavelet basis vectors, and

$b = \begin{bmatrix}b_{x} \\b_{y}\end{bmatrix}$

-   -    are the deformation coefficients.        It is appreciated that it may be desirable in many applications,        to make R_(x) and R_(y) the same for the two patterns f and g,        e.g. R_(x)=R_(y)=R.

The resulting deformation field component vectors 138 and 140 areapplied to the partial derivatives 134 and 136 of the image (e.g., viaan element-wise product function) to provide a deformation component 142of the image with respect to x and a deformation component 144 of theimage g with respect to y. The resulting components 142 and 144 areaggregated by a summing function 146 to derive a deformation component148 of the image g. The deformation component 148 is added to the imageg by a summing function 150 to derive a deformed image {tilde over (g)},indicated at 152. The deformed image 152 thus may be represented as{tilde over (g)}=g+(G _(x) g)∘(Rb _(x))+(G _(y) g)∘(Rb _(y)),  (Eq. 8)where G_(x) and G_(y) are approximate spatial derivative matrices usedin derivative functions 130 and 132 (see also the Notation section).

As described above with respect to Eq. 4, the deformed image {tilde over(g)} may in turn be represented as:{tilde over (g)}=g+D(g)b, where D(g)=[diag(G _(x) g)R diag(G _(y)g)R]  (Eq. 9)In accordance with an aspect of the present invention, the deformedimage {tilde over (g)} may be subtracted from the deformed image {tildeover (f)} by a subtraction function 154 to provide a measure of thedistance 156 between the two deformed images. From Equations 4 and 9,the distance between the two deformed images {tilde over (f)} and {tildeover (g)} may be expressed as

$\begin{matrix}{{{\overset{\sim}{f} - \overset{\sim}{g}} = {f - g + {\lbrack {{D(f)} - {D(g)}} \rbrack\begin{bmatrix}a \\b\end{bmatrix}}}},} & ( {{Eq}.\mspace{14mu} 10} )\end{matrix}$Eq. 10 provides a good indication of the similarity between the images fand g provided that the deformation coefficients a and b are known.Those skilled in the art will understand and appreciate that thedeformation coefficients a and b may be found so as to minimize thedifference in Eq. 10 in a straightforward fashion, such as by quadraticoptimization with respect to the deformation coefficients a and b.Alternatively, an iterative process, such as gradient descent, may beemployed to derive the deformation coefficients. For example, feedback158 and 160 indicative of the distance measure 156 may be applied toiteratively derive the respective coefficients a and b.

In accordance with an aspect of the present invention, the deformationcoefficients a and b may be functionally related to each other. By wayof illustration, assuming that both patterns have identical displacementbasis functions, the deformation coefficients of the image f may be setto be substantially the opposite of the deformation coefficient of theimage g (e.g., a≈−b). By placing such a requirement on the deformationcoefficients, it will be appreciated that the images f and g will tendto deform toward each other, tending to minimize the distance betweenthe resulting deformed images {tilde over (f)} and {tilde over (g)}. Inaddition, the size of the optimization problem may be reduced and theallowable deformations may be further constrained.

In accordance with an aspect of the present invention, one or more costfunctions may be employed to capture correlations between differentpixels in the respective images f and g and/or to favor some deformationfields over others. Referring back to the example of FIG. 3, a costfunction (E1) 160 may be applied to the distance 156 between thedeformed images to derive a first term 162 that provides a weightedapproximate measure of similarity between the images f and g. The costfunction 160, for example, returns a scalar quantity functionallyrelated to the distance 156 and a cost descriptor Ψ. By way of example,Ψ is a diagonal matrix whose non-zero elements contain variances ofappropriate pixels to enable different pixels to have different levelsof importance for purposes of matching. The diagonal matrix Ψ may behand crafted to non-uniformly weight the importance of each pixel sothat more important aspects of a pattern may be weighted more heavily inthe pattern recognition process.

For example, if the images f and g are images of a tree in the wind, thedeformation coefficients should be capable of aligning the main portionsof the image, such as may include the trunk and large branches, whilethe variability in the appearance of the leaves may be captured in thecost term Ψ.

The system 100 may include another cost function (E2) 164 to capture acovariance structure of the deformation coefficients a and b of theallowed deformations. The cost term 164 applies a covariant matrix Γ tothe deformation coefficients a and b to constrain the deformation of theimages f and g. The cost function 164 provides scalar cost quantity 166,which is functionally related to the matrix Γ and the respectivedeformation coefficients a and b (e.g., a and b are inputs to 164). Thecost term 166 (see, e.g., Eq. 11) allows capturing correlations amongcoefficients. For instance, if the major component of a deformation is arotation around the center, then the displacements in x and y directionsare strongly correlated. Similarly, when deforming both images, it islikely that the deformation coefficients for the two images areanti-correlated, e.g., one is deforming in the opposite direction of theother, so that minimal deformation is necessary in both images.

Those familiar with statistics and covariance matrices can easilyconstruct matrices that produce cost terms that are lower when thesetypes of correlations are present in the deformation coefficients.While, in theory, similar correlations can be captured directly in thepixel domain, such as in the cost term involving Ψ, in practice it isdifficult to incorporate such correlations due to the highdimensionality of the data. Consequently, it is easier to use a diagonalmatrix that decouples the effects of different pixels in the appropriatecost term. In contrast, the number of deformation coefficients invectors a and b are small enough to make it possible to use non-diagonalstructures that capture correlations as discussed herein.

Another cost function (E3) 170 may be applied to the respectivedeformation components 120 and 148 to further constrain the elasticdeformation associated with each of the deformation coefficients a andb. For example, a constraint matrix Q also is applied to the costfunction 170 to return a scalar quantities 172 and 174 functionallyrelated to the respective deformation components 120 and 148 and to thematrix Q. The matrix Q captures a global cost of deformation in thepixel space (as opposed to Γ which capture a deformation cost in thewavelet space). For example, a good choice for Q would be a matrixco-linear to Ψ and the scalar ratio between the two matrices wouldreflect the cost of moving {tilde over (f)} and {tilde over (g)} in thetangent planes, with respect to the cost of the distance between {tildeover (f)} and {tilde over (g)}, such as depicted in FIG. 4, with Q=k Ψ.

The cost quantities 162, 166, 168, 172, and 174 are aggregated togetherby a summing function 176 to derive a versatile image distance 178indicative of a measure of the similarity between the images f and g(e.g., the distance there between), which may be expressed as

$\begin{matrix}{{d( {f,g} )}^{2} = {\min\limits_{a,b}{\{ {{( {\overset{\sim}{f} - \overset{\sim}{g}} )^{\prime}{\Psi^{- 1}( {\overset{\sim}{f} - \overset{\sim}{g}} )}} + {\lbrack {a^{\prime}b^{\prime}} \rbrack{\Gamma^{- 1}\begin{bmatrix}a \\b\end{bmatrix}}} + {( {{D(f)}a} )^{\prime}{Q^{- 1}( {{D(f)}a} )}} + {( {{D(g)}b} )^{\prime}{Q^{- 1}( {{D(g)}b} )}}} \}.}}} & ( {{Eq}.\mspace{14mu} 11} )\end{matrix}$Those skilled in the art will understand and appreciate that Eq. 11 isquadratic in the deformation coefficients a and b, which, as mentionedabove, enables the coefficients to be determined by straightforwardmethods when such cost terms are employed.

It thus may be shown that d(f g) may be computed by solving a linearsystem, in which (a, b) are the unknown parameters, and e(a, b) is thequantity to be minimized. Since d(f, g) is at a minimum,

${\frac{\partial{e( {a,b} )}}{\partial a_{x}} = 0},{\frac{\partial{e( {a,b} )}}{\partial a_{y}} = 0},{\frac{\partial{e( {a,b} )}}{\partial b_{x}} = 0},{\frac{\partial{e( {a,b} )}}{\partial b_{y}} = 0.}$Because e(a, b) is quadratic in a and b, setting its derivative to 0results in a linear system that can be solved with standard methods.

It is to be understood and appreciated that other methods may beemployed to solve for the deformation coefficients a and b. For example,as indicated in FIG. 3, the versatile image distance may be fed back aspart of a feedback loop that may be utilized to derive the deformationcoefficients as part of an iterative process, in which the coefficientsare further refined with each iteration. The iterative process may beconsidered more manageable in certain circumstances, such as when thelinear system becomes too large to solve directly.

FIG. 4 is a graphical representation of patterns f and g and theirrespective deformed patterns {tilde over (f)} and {tilde over (g)},illustrating the relationship between the patterns in accordance with anaspect of the present invention. In this example, the elastic distancebetween the patterns f and g is modeled according to a spring systemwith an assumption that Γ⁻¹=0 in Eq. 11, that Ψ is the identity, andthat Q=k Ψ,. That is, each of the respective deformed patterns {tildeover (f)} and {tilde over (g)} is able to move along a linear plane, butthey are attached to each other by a spring of constant elasticity(e.g., an elasticity of 1) and by springs of elasticity k to theirrespective patterns f and g. That is, the deformed pattern {tilde over(f)} is functionally related to the pattern f by a spring constant k andis free to move along a linear plane relative to the pattern f. Themovement along the linear plane for f may be expressed asl _(f) =D(f)a  (Eq. 14)Similarly, the point the deformed pattern {tilde over (g)} is free tomove along its linear plane relative to the pattern g, as may beexpressed asl _(g) =D(g)b  (Eq. 15)At equilibrium, the energy of the system of FIG. 4 may be expressed asd(f,g)² =k(l _(f) ² +l _(g) ²)+({tilde over (f)}−{tilde over(g)})²  (Eq. 16)It is to be appreciated that if Γ⁻¹≠0, certain dimensions may beprivileged within each linear space.

In FIG. 4, for purposes of simplicity of explanation, the linear spacehas one dimension, or only one direction. It is to be understood andappreciated that a system may be implemented in accordance with anaspect of the present invention having multiple dimensions, each ofwhich may be privileged by configuring an appropriate weighting matrixΓ⁻¹.

In order to provide additional context for the various aspects of thepresent invention, FIG. 5 and the following discussion are intended toprovide a brief, general description of a suitable computing environment200 in which the various aspects of the present invention may beimplemented. While the invention has been described above in the generalcontext of computer-executable instructions of a computer program thatruns on a local computer and/or remote computer, those skilled in theart will recognize that the invention also may be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks or implement particular abstract data types.Moreover, those skilled in the art will appreciate that the inventivemethods may be practiced with other computer system configurations,including single-processor or multiprocessor computer systems,minicomputers, mainframe computers, as well as personal computers,hand-held computing devices, microprocessor-based or programmableconsumer electronics, and the like, each of which may be operativelycoupled to one or more associated devices. The illustrated aspects ofthe invention may also be practiced in distributed computingenvironments where certain tasks are performed by remote processingdevices that are linked through a communications network. However, some,if not all, aspects of the invention may be practiced on stand-alonecomputers. In a distributed computing environment, program modules maybe located in both local and remote memory storage devices.

With reference to FIG. 5, an exemplary system environment 200 forimplementing the various aspects of the invention includes aconventional computer 202, including a processing unit 204, a systemmemory 206, and a system bus 208 that couples various system componentsincluding the system memory to the processing unit 204. The processingunit 204 may be any of various commercially available processors. Dualmicroprocessors and other multi-processor architectures also may be usedas the processing unit 204.

The system bus 208 may be any of several types of bus structureincluding a memory bus or memory controller, a peripheral bus, and alocal bus using any of a variety of conventional bus architectures suchas PCI, VESA, Microchannel, ISA, and EISA, to name a few. The systemmemory includes read only memory (ROM) 210 and random access memory(RAM) 212. A basic input/output system (BIOS), containing the basicroutines that help to transfer information between elements within thecomputer 20, such as during start-up, is stored in ROM 210.

The computer 20 also may include, for example, a hard disk drive 214, amagnetic disk drive 216, e.g., to read from or write to a removable disk218, and an optical disk drive 220, e.g., for reading a CD-ROM disk 222or to read from or write to other optical media. The hard disk drive214, magnetic disk drive 216, and optical disk drive 220 are connectedto the system bus 208 by a hard disk drive interface 224, a magneticdisk drive interface 226, and an optical drive interface 228,respectively. The drives and their associated computer-readable mediaprovide nonvolatile storage of data, data structures,computer-executable instructions, etc. for the computer 20. Although thedescription of computer-readable media above refers to a hard disk, aremovable magnetic disk and a CD, it should be appreciated by thoseskilled in the art that other types of media which are readable by acomputer, such as magnetic cassettes, flash memory cards, digital videodisks, Bernoulli cartridges, and the like, may also be used in theexemplary operating environment 200, and further that any such media maycontain computer-executable instructions for performing the methods ofthe present invention and string pattern-related data.

A number of program modules may be stored in the drives and RAM 212,including an operating system 230, one or more application programs 232,other program modules 234, and program data 236. The operating system230 in the illustrated with any operating system or combinations ofoperating systems.

A user may enter commands and information into the computer 202 throughone or more user input devices, such as a keyboard 238 and a pointingdevice (e.g., a mouse 240). Other input devices (not shown) may includea microphone, a joystick, a game pad, a satellite dish, a scanner, orthe like. These and other input devices are often connected to theprocessing unit 204 through a serial port interface 242 that is coupledto the system bus 208, but may be connected by other interfaces, such asa parallel port, a game port or a universal serial bus (USB). A monitor244 or other type of display device is also connected to the system bus208 via an interface, such as a video adapter 246. In addition to themonitor, a computer typically includes other peripheral output devices(not shown), such as speakers, printers etc.

The computer 202 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer260. The remote computer 260 may be a workstation, a server computer, arouter, a peer device or other common network node, and typicallyincludes many or all of the elements described relative to the computer202, although, for purposes of brevity, only a memory storage device 262is illustrated in FIG. 5. The logical connections depicted in FIG. 5include a local area network (LAN) 264 and a wide area network (WAN)266. Such networking environments are commonplace in offices,enterprise-wide computer networks, intranets and the Internet.

When used in a LAN networking environment, the computer 202 is connectedto the local network 264 through a network interface or adapter 268.When used in a WAN networking environment, the computer 202 typicallyincludes a modem 266, or is connected to a communications server on theLAN, or has other means for establishing communications over the WAN266, such as the Internet. The modem 266, which may be internal orexternal, is connected to the system bus 208 via the serial portinterface 242. In a networked environment, program modules depictedrelative to the computer 202, or portions thereof, may be stored in theremote memory storage device 262. It will be appreciated that thenetwork connections shown are exemplary and other means of establishinga communications link between the computers 202 and 260 may be used.

In accordance with the practices of persons skilled in the art ofcomputer programming, the present invention has been described withreference to acts and symbolic representations of operations that areperformed by a computer, such as the computer 202 or remote computer260, unless otherwise indicated. Such acts and operations are sometimesreferred to as being computer-executed. It will be appreciated that theacts and symbolically represented operations include the manipulation bythe processing unit 204 of electrical signals representing data bitswhich causes a resulting transformation or reduction of the electricalsignal representation, and the maintenance of data bits at memorylocations in the memory system (including the system memory 206, harddrive 214, floppy disks 218, CD-ROM 222, and shared storage system 210)to thereby reconfigure or otherwise alter the computer system'soperation, as well as other processing of signals. The memory locationswhere such data bits are maintained are physical locations that haveparticular electrical, magnetic, or optical properties corresponding tothe data bits.

In view of the exemplary operating environments and functional systemshown and described above, a methodology, which may be implemented inaccordance with the present invention at a computer, will be betterappreciated with reference to the flow diagrams of FIGS. 6A and 6B.While, for purposes of explanation, the methodology is shown anddescribed as a series of steps with respect to the flow diagrams, it isto be understood and appreciated that the present invention is notlimited by the order of steps, as some steps may, in accordance with thepresent invention, occur in different orders and/or concurrently withother steps from that shown and described herein. Moreover, not allillustrated steps may be required to implement a methodology inaccordance with the present invention.

FIGS. 6A and 6B are flow diagrams respectively illustrating amethodology for determining deformation coefficients and computing ameasure of similarity between patterns in accordance with an aspect ofthe present invention. The methodology begins at step 300 in whichpatterns f and g are provided for which a matching process, inaccordance with the present invention, is to be implemented. Thepatterns may be any type of patterns, such as, for example, images(e.g., handwriting, alphanumeric characters, finger prints, etc.), audiopatterns (speech, marine animal sounds, etc.). The process then proceedsto step 310. Furthermore, the patterns provided at step 300 may besmoothed versions of the original patterns. Smoothing may be necessaryin some cases where the data is very noisy because in further steps,derivatives will be computed, and derivative are sensitive to noisydata. For instance, smoothing can be obtained by convolving the originalpatterns with a Gaussian function of standard deviation σ.

At step 310, a derivative of each pattern f and/or g being deformed istaken. For example, a spatial derivative may be taken of the pattern fwith respect to x and y, such as partial derivatives for each dimensionassociated with the pattern. At step 320, a determination is made as towhether the deformation coefficients are known. If the coefficients areknown, the process proceeds through connector A to FIG. 6B. However, ifthe deformation coefficients are either not known or it is desirable torefine the coefficients, the process proceeds to step 330.

At step 330, a substantially smooth deformation field is applied to(e.g., multiplied with) the derivative of one or more of the patterns.In accordance with an aspect of the present invention, the deformationfield is a smooth and generally non-uniform field parameterized byfunctional basis vectors and associated deformation coefficients. Thefunctional basis vectors may be linear transformation matrices, such as,for example, wavelet basis vectors, Fourier type basis vectors, sin orcosine vectors, or other suitable basis vectors. The basis vectors aremultiplied by the deformation coefficients, such as by employing anelement-wise product function. At this stage of the process, thedeformation coefficients are unknown parameters and, thus, may becharacterized as variables. The deformation field is, in turn applied tothe spatial derivative(s) via an element-wise product function toprovide a corresponding deformation component for each pattern beingdeformed. As mentioned above, it may not be necessary to deform bothpatterns.

From step 330, the process proceeds to step 340 in which eachdeformation component of is added to its respective original pattern fand/or g, resulting in a deformed pattern, such as may be represented byEq. 3 noted above. It is to be appreciated that the deformationcomponent is decoupled from the image f. As a result, a particulardeformation field may be easily applied to other patterns to deform suchother patterns in a corresponding way.

When both patterns have been deformed, each deformed pattern f and g hasdeformation coefficients a and b, respectively. The deformationcoefficients a for the pattern f may be functionally related to thedeformation coefficients b for the pattern g. For example, assuming thatboth patterns have identical displacement functions, it may be desirableto cause a to tend toward −b (e.g., a≈−b). As a result of suchrelationship between coefficients, the patterns will tend to deformtoward each other. Additionally, the size of the optimization problemmay be reduced and the allowable deformations may be further constrainedby the relationship between deformation coefficients.

Next, the process proceeds to step 350 in which an expression for asimplified distance between the deformed patterns (e.g., an error image)is derived. The distance, which may be expressed as the differencebetween the two deformed patterns, is a linear system, such as set forthabove with respect to Eq. 10. An indication of the distance between thepatterns is derived by subtracting one pattern from the other.

From step 350, the process proceeds to step 360 in which deformationcoefficients are computed by minimizing the cost associated withdeforming the respective images. The cost minimization balances theerror image (e.g., simple distance) of the patterns f and g with theassociated deformation provided by the respective deformationcoefficients.

As mentioned above, a cost function may capture correlations betweendifferent parts in the respective patterns f and g. The cost function,for example, may include a diagonal matrix whose non-zero elementscontain variances of appropriate pixels. The cost function may operateto provide different pixels with different levels of importance so thatmore important aspects of a pattern may be weighted more heavily in thepattern recognition process. Another cost function, which varies as afunction of the deformation coefficients a and b, may capture acovariant structure associated with the deformation coefficients of theallowed deformations to constrain the deformation of the images f and g.Yet another cost function, which varies as a function of the respectivedeformation components (derived at step 330), may further constrain theelastic deformation associated with each of the deformation coefficientsa and b as applied to the patterns f and g. Each cost function mayreturn a scalar quantity of the form x^(T)Mx; where M is a matrix ofvalues, x is a quantity to which the cost function is applied, and ^(T)denotes a transpose operation. As a result, after application of suchcost functions, the aggregate expression provides an indication of thedistance between the patterns f and g.

For example, the deformation coefficients may be computed by quadraticoptimization by minimizing the cost of deformation between the patternsf and g with respect to the deformation coefficients a and b.Alternatively, the deformation coefficients may be solved by aniterative process, such as gradient descent. Those skilled in the willunderstand and appreciate that other methods also may be employed toderive the deformation coefficients in accordance with an aspect of thepresent invention.

After the deformation coefficients have been computed in step 360, theprocess proceeds to the connector A to compute a measure of similaritybetween the patterns f and g as shown in FIG. 6B. The methodology shownin FIG. 6B is similar to that shown and described with respect to steps330-360 of FIG. 6A. Briefly stated, the process continues to step 400,in which a substantially smooth deformation field is applied to (e.g.,multiplied with) the spatial derivative of the pattern(s). In accordancewith an aspect of the present invention, the deformation field is asmooth and generally non-uniform field parameterized by functional basisvectors and the deformation coefficients determined at step 360. Thefunctional basis vectors may be linear transformation matrices, such as,for example, wavelet basis vectors, Fourier type basis vectors, sin orcosine vectors, or other suitable basis vectors. The basis vectors aremultiplied by the deformation coefficients, such as by employing anelement-wise product function. The deformation field is, in turn appliedto the derivative(s), previously determined at step 310 via anelement-wise product function to provide a corresponding deformationcomponent for each pattern being deformed. In accordance with an aspectof the present invention, both patterns f and g may or may not bedeformed at step 400.

From step 400, the process proceeds to step 410 in which eachdeformation component of is added to its respective original pattern fand/or g, resulting in a deformed pattern (see, e.g., Eq. 3). When bothpatterns have been deformed, each deformed pattern f and g hasrespective coefficients a and b, which may be functionally related toeach other.

Next, the process proceeds to step 420 to derive an indication of adistance between the patterns. The distance, which may be expressed asthe difference between the two deformed patterns, is a linear system,such as set forth above with respect to Eq. 10. From step 420, theprocess proceeds to step 430 in which a cost of deformation among thepatterns f and g is evaluated to determine the similarity between thepatterns. At this stage, it is assumed that the parameters are known.For example, the measure of similarity may be referred to as an elasticdistance between the patterns f and g, which may be evaluated bysubstituting values of the deformation coefficients a and b into thedistance expression, such as set forth above with respect to Eq. 11 orEq. 12. The cost of deformation may thus be evaluated to determine thelikelihood of a match between the patterns f and g. The deformation costfunction may be substantially identical to that described herein.

FIG. 7 illustrates several examples of a deformed image patternresulting from application of eight randomly selected deformation fieldsto an image pattern in accordance with an aspect of the presentinvention. In particular, the deformation coefficients have beenselected from a unit-covariance Gaussian to provide respectivedeformation fields that are substantially smooth. The resultingdeformation fields, in turn, were applied to an image pattern of adigit.

FIG. 8 illustrates an example in which deformation fields have beenestimated for matching two images 450 and 452 of a face of the sameperson but with different facial expressions. In this example,deformation fields 454 and 456 have been derived from application of thealgorithm to the image patterns 400 and 402, respectively. Thedeformation fields 454 and 456 operate on the image patterns 450 and 452to produce respective deformed images 458 and 460. In this case, the Ψmatrix has been set to identity and the matrix Γ has been set by hand toallow a couple of pixels of deformations. As a result of applying thedeformation fields 454 and 456, the image patterns 450 and 454 tend todeform toward each other, as shown by deformed images 458 and 460, sothat a more accurate measure of similarity between the patterns may bedetermined, such as described herein. Those skilled in the art willunderstand and appreciate that, in accordance with teachings containedherein, other acceptable (or even better) deformation fields may beobtained, such as by minimizing the cost parameters associated with Eq.11. The resulting deformation fields may, in turn, be employed tofacilitate measuring similarity between patterns in accordance with anaspect of the present invention. What has been described above includesexamples of the present invention. It is, of course, not possible todescribe every conceivable combination of components or methodologiesfor purposes of describing the present invention, but one of ordinaryskill in the art will recognize that many further combinations andpermutations of the present invention are possible. Accordingly, thepresent invention is intended to embrace all such alterations,modifications and variations that fall within the spirit and scope ofthe appended claims. Furthermore, to the extent that the term “includes”and variants thereof or the term “having” and variants thereof are usedin either the detailed description or the claims, such terms areintended to be inclusive in a manner similar to the term “comprising.”

1. A computer-implemented method to facilitate pattern recognitionbetween a first pattern and a second pattern, comprising: deforming thefirst pattern and the second pattern to derive deformation componentsfor the first pattern and the second pattern; aggregating the firstpattern and the deformation component thereof to define a first deformedpattern; aggregating the second pattern and the deformation componentthereof to define a second deformed pattern; and determining a distancebetween the first deformed pattern and the second deformed pattern. 2.The method of claim 1, the deformation component of the first patternand the deformation component of the second pattern are respectivelyderived from applying a deformation field having associated deformationcoefficients to a derivative of the first pattern and a derivative ofthe second pattern.
 3. The method of claim 2, further comprisingdetermining a linear approximation of the first deformed pattern and alinear approximation of the second deformed pattern.
 4. The method ofclaim 3, the linear approximations are determined based at least in parton the deformation field.
 5. The method of claim 3, further comprisingcomputing a distance between the first deformed pattern and the seconddeformed pattern based at least in part upon the linear approximation ofthe first deformed pattern and the linear approximation of the seconddeformed pattern, the distance is used to determine whether the firstand second patterns match.
 6. The method of claim 5, further comprisingapplying a cost function to the computed distance to capturecorrelations between the first and second patterns.
 7. The method ofclaim 6, the cost function comprises at least one of a variant structureand a covariant structure related to the deformation coefficients. 8.The method of claim 2, the deformation field is parameterized byfunctional basis vectors and deformation coefficients.
 9. The method ofclaim 8, the functional basis vectors are derived from at least one ofan inverse wavelet transform and a fast Fourier transform.
 10. Acomputer-implemented system that facilitates pattern recognition betweena plurality of patterns, comprising: a deformation function componentthat derives deformation components for a first and a second pattern bydeforming the first and second patterns; an aggregator that aggregatesthe first and second patterns and the deformation components thereof todefine respective deformed patterns; and a minimization component thatminimizes a distance between the respective deformed patterns.
 11. Thesystem of claim 10, the deformation components are derived from applyinga deformation field having associated deformation coefficients to aderivative of the first and second patterns.
 12. The system of claim 11,further comprising a component to determine linear approximations forthe first and second deformed patterns.
 13. The system of claim 12, thelinear approximations are determined based at least in part on thedeformation field.
 14. The system of claim 12, further comprising adistance computing component that computes a distance between the firstand second deformed patterns based at least in part upon the linearapproximations of the first and second deformed patterns.
 15. The systemof claim 14, further comprising a cost computing component that appliesa cost function to the computed distance to capture correlations betweenthe first and second patterns.
 16. The system of claim 14, furthercomprising a matching component that determines whether the first andsecond patterns are a matched based at least in part upon the computeddistance.
 17. The system of claim 11, the deformation field isparameterized by functional basis vectors and deformation coefficients.18. The system of claim 17, the functional basis vectors are derivedfrom at least one of an inverse wavelet transform and a fast Fouriertransform.
 19. A system that facilitates pattern recognition between atleast two patterns, comprising: means for deriving a deformationcomponent for a first pattern; means for deriving a deformationcomponent for a second pattern; means for aggregating the first andsecond patterns and their respective deformation components to definerespective deformed patterns; and means for determining a distancebetween the respective deformed patterns.
 20. The system of claim 19,further comprising means for determining the distance at least in partby comparing computed linear approximations of the respective deformedpatterns.